Periodic Orbits in the Problem of Three Bodies with Repulsive and Attractive Forces

Periodic Orbits in the Problem of Three Bodies with Repulsive and Attractive ForcesAuthor(s): Daniel BuchananSource: American Journal of Mathematics, Vol. 52, No. 4 (Oct., 1930), pp. 899-913Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2370723 .

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Periodic Orbits in the Problem of Three Bodies with Repulsive and Attractive Forces.

BY DANIEL BUCHANAN.

1. Introduction. This paper deals with periodic orbits described by two mutually repellant infinitesimal bodies which are attracted by a finite body. The forces of repulsion and attraction are assumed to vary according to the Newtonian law of the inverse square. Two types of periodic orbits for this system were obtained by Rawles.* In the first type, which will be here designated as the circular orbits, the repellant particles move in equal circles the planes of which are parallel. The line joining the centres of these circles is normal to their planes and is bisected by the centre of gravity of the finite body. The particles remain on the same generating line of the cylinder through these circles.

In the orbits of the second type, here designated as the arc orbits, the three bodies remain in the same plane. The infinitesimal bodies oscillate in arcs of curves, which are symmetrically situated with respect to the finite body. Langmuir t first calculated these orbits by numerical integration and they are also discussed by Van Vleck.t

The problem considered in the present paper deals with periodic oscillations in the vicinity of the circular orbits. Only the construction of these orbits is made but the convergence of the solutions obtained is assured by a theorem due to MacMillan.-? The author begs to acknowledge the assistance of Mr. H. D. Smith, M. A.,? in checking certain algebraic expressions in the construction and in making the computation for the numerical examples.

Second genus orbits in the vicinity of the arc orbits have also been obtained by the author but they are discussed in another article. 1

* Rawles, " Two Classes of Periodic Orbits with Repelling Forces," Bulletin of the American Ma7thematical Society, Vol. 34, No. 5 (1928), pp. 618-630.

t Langmuir, Physical Review, Vol. 17 (1921), pp. 339-353. $ Van Vleck, "Quantum Principles and Line Spectra," Bulletinr of the National

Research Council, Vol. 10, Part 4, No. 54, p. 89. ? MacMillan, Transactions of the American Mathematical Society, Vol. 13, No. 2,

pp. 146-158. ? Smith, A thesis submitted in the Department of Mathematics for the degree of

M. A. in the University of British Columbia. Buchanan, " Second Genus Orbits for the Helium Atom," Transactions of the

Royal Society of Canada, Third Series, Vol. 23, Sec. 3 (1929), pp. 227-245. 899



900 BUCHANAN: Periodic Orbits in the Problem of

As there is a similarity between the three bodies in this problem and the helium atom, we shall refer to the finite body as the nucleus and to the par- tidles as electrons. No use, however, is made of the quantum mechanies nor of Larmor's theorem.*

2. The Circular Orbits. The units of time and space will be chosen so that the gravitational constant of attraction is unity. Let k2 denote the ratio of the repulsion to the attraction. Then the force function of the system is

U = 1/p1 + 1/p2-k2/A

where p1 and P2 are the distances between the electrons and the nucleus, and A is the distance between the electrons. If we take a system of rectangular coiirdinates with the origin at the nucleus and denote the coordinates of the electrons as (xj, yj, z;), (j - 1, 2), then the differential equations defining their motion are

xj= U/aaxj, y j= u zayj f zj"-aulzzj, (1) pj2 Xj2 + yj2 +Z2, (j=1,2),

A2 (Xl- Xa2)2 + (yl -y2)2 + ZIL (z-2) 2.

When the restrictions

(2) X1 - X2, YL -Y2, Z1 Z2

are made, as in Rawles' paper, the differential equations become

X - X/p3 + k2/4x2, (3) y"=-y/p3

z - z/p3,

where the subscripts 1 or 2 have been dropped. These equations possess the integrals

?/2 (x'2 + y'2 + Z'2) i J/p - k2/4x + const., y'z - yz= const.

The solutions of the differential equations are

X= (k2/4)/3 -m, say, (5) y= (1 _ M2)'/2 sin (t- t),

zc e m2)ci a li COS (t a t h)

which are the circular solutions obtained by Rawles. They denote the circles

* Larmor, Phitlosophical Magazzine, V, Vol. 44 (1897), p. 503; Richardson, The Electron Theory of Matter (1916), p. 258.



Three Bodies with Repulsive and Attractive Forces. 901

with centres at (?+ m, 0, 0), radii (1 - m2)V'2 and whose planes are parallel to the yz-plane. The electrons rotate in these orbits from the positive z-axis to the positive y-axis. If the solutions are to be real, m2 cannot exceed unity. When m2 - 1, however, the solutions reduce to point circles but this simple case will be excluded from our consideration.

We shall refer only to the one circle, viz., that having its centre at (m,O,0o).

Orbits of Three Dimensions.

3. The Differential Equations. Let the motion be referred to a system of rotating axes x, -, t. The x-axis remains unchanged while the v-$-axes rotate in the yz-plane in the direction in which the electrons move and with their angular velocity. Further, let -y, z at t to. Then the necessary transformations are

(6) y -q Cos (t to) + esin (t to), z _= =sin (t -to) + 4 cos (t- to),

and the differential equations of motion (3) become

x1 XI,o + 7. /4x',

(7) X)+ 2$' -p' /p3,

A particular solution of these equations is

(8) x =m, v 0, =- (1 _mA2) ?,

which are the equations of the circular orbit with respect to the rotating axes. In order to determine deviations from the circular orbit, let

x = m + yp,

(9) q

O + yq, (= (1-m2)1/2+yr,

t -to - ( 1 + 8) /2,r

where

p, q, r are new dependent variables, y is a parameter representing the scale factor of the new orbits, 8 is a constant depending upon y, r is the new independent variable.

When equations (9) are substituted in (7) and the factor y is divided out,



902 BUCHANAN: Periodic Orbits in the Problemr of

the following differential equations are found, the dots denoting derivation with respect to T;

p + 3(1 + 8) (1-M2)p -3(1 + 8) M(1 _M2)'/2r (1+ 8) [7P2 + y2P3 + + yjPj+l + * * ],

(10) q + 2 (1 + 8):/2r = 1+ 8) [YQ2 + **+ yjQj+l *] r- 2(1 + 8)Y12q- 3(1 _ M2 r1 qn)- 3 (1 + 8) M (1 - 2)1/2p

= (1 + 8) [yR2 + * + yjRj+i + * * *],

where Pj, Qj, Rj (j 2, 3, . * ) are polynomials in p, q, r of degree j. In Pj and Rj, q enters to even degrees only, while in Qj it enters to odd degrees only. So far as the computation has been carried out we have

P2 3(1/mr + 3m/2 - 5m3/2) p2 + 3mq2/2

-3m(2 - 5m2/2)r2 +3(1 m2)'/2(1_5m2)pr,

P3= (3/2 -15m2+35m4/2)pl3 + (15m/2)'(1 _M2)'/2(7m2 3)p2r

+ (3/2) (1 -5M2) pq2- 3 (2- 35M2/2 + 3 5M4/2)pr2 -(15m/2)(1 M2) 1/2q2r + 5 ( 2-7m2/2) r3,

Q2 = 3mpq + 3(1 M2)?qr, Q3 =(3/2) (1 5m2)p2q -15m(1l m2) /2pqr + 3q3/2

-3 (2 - 5m2/2)r2q,

R2 ==(3/2) (1 m2)1/2(1 - 5m2)p2 - 3m(4 - 5m2)pr

+(3/2) (1 m2)/2q2 -3(1 m2)a/(- 5m2/2)r2, R13 =(5m/2) (1 m2)1/2 (7m2 -3)p3

+ 3[1/2 + 15m2 - 35m4/2 - (5m/2) (1 - m2)l/2]p2r

- (15m/2) (1 m2)12pq2 + 15m(1m2)l/2(2-7m2/2)pr2

+ 3[1/2 - 5M(l m2) 1/2]q2r

-[27/2- 15m2 (35/2) (1 _M2)5/2']r3.

On integrating (10, b) we obtain

(11) q 2 (1 + 8)/2r + C

+ (1 + 8) (yQ2 + * *+yjQj+l * . w )dT

where C is the constant of integration. As q and r are later developed as power series in y we shall put

(12) C ( C0() + CO(l1) + + COiy, +

When the substitutions are made for C in (11) and for q in (10, c) we obtain, on repeating (10, a) and (11) for reference,




3(1 + 8) (1 -m2)p -3(1 + 8)m(1 m2)?2r 00

(1 +8) Y yP j+i, j=l

s , ~~~~~00 (13) q -2(1 + 8)1/2r + (1 + 8) Y. -y'Qj+dT + I Ci(')yj,

j=l j=O r + (1 + 8).(1 + 3m2)r- 3(1 ? 8)m41 - m2)1/2p

00 00

(1+8) Y yjRj+l +2(1+ 8)Y2 Y Ci(J)yJ j=1 j=o

00

+ 2 (1 + 8)3/2 Y. yJQj+,dT. j=1

We shall now take (13) as the three defining equations for p, q, r.

4. The Equations of Variation and their Solutions. If we consider only the terms of the equations (13) which are independent of y we obtain the equations of variation. They are

p + 3(1 3m(1 - m2)%r 0, (14) q +2r =- C, (?

M22 p C (0). r + (1 + 3m2)r -- 3ml (- m-)p C'.

The first and third equations of (14) are independent of the second and will be considered first. We shall make use of the operator D to denote d/dr. Then (14, a) and (14, c) may be expressed as

[D2 + 3(1-rm2)] p-3m(1l m2)1r- 0, (15) ~ 3m(1 m2)p+ [D2 + 1 +3m2]r 2C,'0).

The functional determinant of these equations is

(16) iZ) | D2+3(1-m2), -3m(1-_m2)'/2 (16) ~~~~-3m(1-M2) 1/2, D2 + 1+3M2 I

=D4 +4D2 + 3(1- m2).

On equating P to zero, as in the method of solving sets of linear differential equations with constant coefficients, we find the roots

D2 - 2 +(1 +3m2)2, _2 (1 +3m2)?.

As m2 must be less than 1 in order that the circular solutions shall be real, both roots for D2 are therefore negative. If we put

-2 4-(1 + 3m2)1/2 -U12, -2- (1 + 3m2)'/2 = 2 2

then D ?-+ iaj, ? iO2,



904 BUCHANAN: Periodic Orbits in the Probleim of

and the complementary-functions of (15) are thus found to be

(17) P = Aiej.6T + A2e 1i,T ? A3eio2T + A4eoriT,

r = B,e061T + B2ei1'T + B3e '2r + B4e-io2T,

where A,, Bj, (j 1 l * ,4) are constants of integration. Only four of these constants are independent as the following relations hold,

Ai wvBj, (j 1,2,3,4; v 1,2), (18) w= 3m(1 - 2) -/[ 3m2 +(1 + 3m2)2],

W2 3m(1 - m2)2/[1 3m'2 -(1 + 3m2)I2].

There are therefore three sets of generating solutions, viz.,

I p = w. (Bie,.61T + B2e~$iT), = B,e06'i + B2e-4ci;

Period = P1 = 27r/u1. II P = 2 (Bae462Tr + B4e-$6f2T),

r = Bseti2T + B4e-iq2T;

Period = P2 = 27r/o-2.

III p = (oi(Be641rT + B2e-ij1T) + W2(B3e, 2T + B4e-i2T), r = BleilT + B2e "1T + B3eiT2,T + B4eri2T;

Period = P3 = n2P1 = n1P2.

The last solutions, .III, exist only when a, and U2 are commensurable, i. e., when

911a2 n1/n2,

where ni and n2 are relatively prime integers. Orbits are constructed in the sequel by using only the first two generating

solutions. The construction of orbits having generating solutions III was attempted but: abandoned on account of the complexity of the problem.

5. Outline of the Construction of Periodic Solutions. There is the same construction for orbits having the generating solutions I or II except for the subscripts 1 and 2, respectively, on a- and o.. We shall therefore drop these subscripts and restore them in the final solutions.

We propose to show that p, q, r, 8 can be determined as power series in y so that p, q, r shall be periodic with the period P (= P1 or P2) and shall satisfy certain initial conditions, to be discussed presently. Accordingly we put

00 00

P E pjy', q E qjy1, ( 1 9 ) jo= j=0 (19) ~~ ~~~~00 00

r E rjqJ, 8= E iyi. j=o j=1




Let these substitutions be made in (13) and let the resulting equations be cited as (13'). On equating the coefficients of the various powers of y in (13') we obtain sets of differential equations in pi, qj, rj. We propose to show that these equations can be integrated and that the various Sj and the constants of integration at each step can be determined so that pj, qj, rj shall be periodic and shall satisfy the initial conditions, now to be discussed.

6. The Initial Conditions. It will be observed in the next section that at each step of the integration four arbitrary constants arise which are not determined by the periodicity conditions. We therefore impose four initial conditions. Let us suppose that

p(O)= r(O)= q(O)= 0, r(O)# 0.

As r carries the factor y in (9) we may take r(O) 1 without loss of gen- erality. When these initial conditions are imposed upon (19) we obtain

(20) pj (0) ~ rj (() ) ( qj(O) o, (j 0,1, 2 .

r1(0) 1, rj(0) 0, (j 2, 3, 4, . .

7. Construction of the Solutions.

Terms independent of y. When we equate the coefficients of the terms in (13') which are independent of y we obtain equations which are the same as (14) except for the subscript 0 on p, q and r. The solutions which have the period P1 or P2, except for certain terms in T, are

p = (Bi + B2(o)e-iaT) + 2m(1 - (21) qo =(2i/a) (B 1(0e'" - B2(O)riTT), 3C1(0)T + 02(0)

rO= Bj(?'eaT + B2(0)e-iaT + 2C,(O),

where B and C, here and henceforth, with various subscripts and superscripts are constants of integration.

In order to satisfy the periodicity conditions we must put C()=- 0. When we impose the condition po(0)= 0 we obtain B 0) =- B2 0), and con- sequently the condition ro(0)= 0 is satisfied. Then from q0(0)= 0 we obtain C2'=- 0 and from ro(0)= 1 we have

B, (0) =-- B2 ( 1/2.

The periodic solutions at this step which satisfy the initial conditions then become (22) Po = (o cos Cr, qo = (2/a) sin aT, r0 = cos aT.

Terms in -y. The differential equations arising from the terms in y in (13') are




[D2 + 3 (1 -2] 3m(l M2)lAl pl (23) 3qmn( (1_- M2)1/2pl + [D2 + 1 + 3m2]rl - RK + 2C,('),

q1= 2r + C(l) +5 dr, where

p(l) ao(1) + 8al(I-) cos ri- + a2(1) COS 2rT,

Q(1) 8ib,(') sin ar + b2(') sin 2arYT

R(1) CO(1) + SlCl(l) COS Tr + C2(1) COS 2rT; ao(=) (3/2) (1/m + 3m/2 5m3/2) W2 + (3/2) (1 M2)1/2(1 5m2)

-3m(l - 1/U2 - 5m2/4),

a1l') =-3(1 -M2>) o + 3m(1 _m2)1/2,

a2C1) (3/2) (1/m- 3m/2 -5m3/2)O2 + (3/2) (1 M2)1/2(1 5m2M

-3m(l + 1/a2 -5m2/4),

bi(l) , 1 b2(1) - ( 3/or) [mw+ (1 _M2)1/2],

1) (3/4) (1 M2)1/2(l_5M2)W2 3m(2 M2/2) - 3(1 - 2 )m 2 (1/2 - 1/o2 5M2/4),

Cl=C) 3m (1 - M2)1/2 + (1- 3M2),

C2(1) (3/4) (1 m_2)1/2(1 _5M2) ,20 3m (2 _ 5M2/2)w - (3/2) (1 _ M2)

l (1+ 11a2 -5m2/2).

The solutions of (23, a and b) will be considered first as (23, c) depends upon r1. The complementary functions of (23 a) and (23 b) are

(24) Pi - (B1) ei- + B2(l)eiT-) + 2m(1 M2) 1ACi(), () rl = Bjl')ei-rr + B2('1e-i- + 2C,(1).

The particular integrals of pl and ri, expressed symbolically, are

[D2 + 1 + 3m2]P(_) + 3m(1 m2)1/2R(l)

(25) 1 D4 + 4D2 + 3(1 -r2) ( 3m(1- m2)1/2P() + [D2+ 3(1 - M2)]R(1) ri -D D4+ 4D2 + 3(1 m2)

In order that pl and r, shall be periodic the coefficients of cos\ST in the numerators of the above expressions must vanish, inasmuch as - o2 is a root of the denominators. Hence

6 1[a, I){-o-2 + 1 + 3m2} + c1(l){3m (1- _M2)V2}] 0,

(6 [a1(l){3m(1 m2)1/2} + c (1){02 + 3(1 -M2)] 0.

The functional determinant of 81a1(1) and 81c1i1) in the above equations is

or4- 4cr2 4 3(1 m2),




and this vanishes as - U2 is a root of !D in (16). Therefore the two equations in (26) are equivalent. They are satisfied only by 8& 0. The particular integrals then become

(27) pi o(l + cc2(1' cos 2O'T, r1 yo(1) + 72(1) cos 2aT,

where 1 + 3M2 + m

?o 3(1 m2) aO + 0_ M2)1/2CO

(1 -4 U2 + 3m2)a2(1) + 3m(1 M2)1/2c2(l)_ 22 16a'4- 16 U2 + 3(1 -M2)

(1) = - M2)1/2ao co

y2(1) = 3m(1 _2) /a2(l) - {4 U2 3(1 M2)}C (-)

16U4- 16 U2 + 3t(1 M2)

When (24) and (27) are combined we obtain the complete solutions for p, and r1.

The third equation of (23) can now be integrated, the integral 'being

q- (2i/o) (Bj1e B2(1e ia-r) - (3c1a') + 2yo(1))r + /2(1) sin 2YT + C2(1),

where -2(l) (3/4cr3) [mo + (1 _ m2)1/2 - 2y2(1)]

On applying the periodicity and initial conditions to the complete solutions for pl, q1, r1 we obtain

C0'(' = - (2/3)yo(l)o 2() C2 O, B ='l' =B2( )-(1/6)yo(l) - (1/2)y2(').

The desired solutions at this step are thus found to be

p= FO(l + F1(') cos aT + F2(') cos 2OT,

(28) q1 = G101) sin rT + G2(1) sin 2aTr,

ri =HO") ? Hl( COS cri + HU2"1 cos 2aT,

where O(I) = 0f(l) (4m/3) (L m2 -12y(I)

F1(l) = 2BI(I)*1,, F2(1) =2

GI= -(4/o)Bi(1), G2'' 2 H[o") = (113)-yo('). Hj.(1) --= 2BI'1', H2(1 ' Y2'l

Terms i,n y2. It will be necessary to consider the terms in y2 in (13') before the induction to the general term can be made. These terms are



908 BUCHANAN: Periodic Orbits in- the Problemn of

[D2 + 3(1 -M2)] p2-3m( 1J M2)Y2r2 p(2)

(29) 3mn(l - -2)?p2 + (D2 + 1 + 34n2)r2 - R2' + 2C (2),

q2= 2r2 + f Q(2'dr 82ro + C1,(2,

where p(2) ao(2) +(82d1(Z) + a('2) COS Cr

+ a2 (2) cos 20 r + a3 2) cos 3CrT,

(2)= co(2) + (82d2 (2) + C- (2) ) COS (IT

+ C2 (2) COS 2(r + C3 (2) cos 3car, Q (2) = b_(-) sin Sr + b2'2) sin 2orr + b3 (21 sin 3aTr, di (2) 3?? (1- _M2)12 -3 (1- _2) W,

d2(2) = - 3m2+3m(l _M2)1%W.

The values of the various a's, b's, and c's were computed by Mr. Smith but his results are omitted here.

The complementary functions and the particular integrals of the fiTst two equations of (29) are the same as (24) and (25), respectively, with the appropriate changes in subscripts and superscripts. The equations similar to (26) which must be satisfied in order that the particular integrals for pl and r1 shall be periodic, are

30) (1-a2 + 3mA2) [82d1(2) +a1(2)] + 3m(1 -M2)/2 [82d2(2) + c(2)] =0O, 3m(1 - 2)W) [82d1(2) + a1(2)] + {- U2 + 3(1 - M2)} [E2d2(2) + Ci(2)] O

The determinant of the coefficients of the expressions in the brackets [ ] is the same here as in (26) and therefore vanishes. Hence the above equations are identical and can be satisfied by a proper choice of the single arbitrary 8. The required value of 82 iS

(f2 _2 37)a(2) 2)n( /2e (2)2 (31) 8 (31) 82 -

(1- o2 + 3md2) 1d(2) + 3m (1 - mrz) 2d2(2)

When 82 is thus determined, the complete solutions for P2 and r2 will be periodic and will have the form

P2 = w(B1(2)ei'-r + B2(2)e-ia)+ 2m(1 - m2)-2C1(2)

32 ~~~+ ao (2) + (2) CO S (2) CS3T (32) + c +2 ? 2 cos 2oar + a3 cos 3(IT, (32)

r2 B (2e 5 + B2(2e-iar + 2C,(2)

+ yo(2) + y2 (2) cos 2ar + y3(2) cos 37T,

where the a's and y's are linear in the a's and c's. On substituting (32) in (29 c) and integrating we obtain

q2 =(2i/oa) (Bi(2)eo-$-B2(2)e Tr) +(3Ci(2) + 20o(2) )T

+ C2(2) + ,83(2) sin or + I32(2) sin 2(rr + /33(2) sin 3orr,



Three Bodies with Repulsive 'and Attractive Forces. 909

where

=(2) (1//a)S2 (1/o2) bi(2)

92 (2) = ( 1/1ff) /2 (2) - ( 1/4f2)b b2(2)

3(2) (2/3cr)ys(2) -(2/9of) b3)2).

When the periodicity and initial conditions are applied we have

C =(2) -(2/3)7yo(2) C2(2) = 0,

B, (2) B2 (2) =-(1/6)yo (2) -y2 (2) + y3 (2).

The solutions at the third step are therefore

3 V=0

q2 E Gv3) SinfvcTr, v=1

3

r2 HV2) cos Vor, V=O

where Po (2) 2M( 1 m2) -1/2c_(2) + 20(2)

P1(2) 2wB1 (2) Fj (2) = aj(2) (j 2, 3,), G,(= -(4/o)B1(2) + /13(2), G7(2) = j (2), (j= 2, 3), Ho(2) 2c,(2) + yO(2),

HIJ(2) 2B=(2),Hj(2) =yj(2) (j 2, 3).

8. Induction to the General Term. Let us suppose that the pj, qj, rj have all been determined for j = 0, , n - 1 and that they are of the form

j+1 pi Fv(j) Cos VcTr,

V=O

j+1 (33) qj = GvQi) sin vnr,

v=1

j+1 rj N Hv1j) cos5vr, (j= 0, , n -),

v=O

where the Fvi, Gv(j) Hv(j) are functions of m. Further, let us suppose that 8,, * - *, 8.-, have been uniquely determined. We wish to show from these assumptions, from the differential equations, and from the initial and periodicity conditions that pn, qn, rn have the same form as (33) for j = n, and that 8. is a uniquely determined constant.

The differential equations obtained by equating the coefficients of yn in (13') are

[D2 + 3 (1-_ 2) ]ppn-3m(l _ M2) /2r. p(n), (34) -3m(l - Mm2)/2pn + [D2 + 1 + 3m.2]rn (n) + 2C,(n)

qn-- 2rn + QQ n) d- + cl (n) - 8ro




where P = 38. (1 - M2)po + 38.m (1 _ M2) - 2ro

+ terms in pj, qj, rj, 8jS, n) 38nm (1 m2)po -8(1 + 3m2 )ro

+ terms in pj, qj, rj, 8j, Q(n) terms in pj, qj, rj, 8j, (j= *0, , n-i; o0 0).

The undetermined constant 8. enters the right members only where it is expressed and not in the other terms. In p(n) and R(n) the powers of the q's are even while in Q(n) they are odd. Hence p(n) and R(n) are sums of cosines of multiples of or while Q (n) is a sum of sines of multiples of qr. They have the form

P(n) ao(_) + (d1(n)8n + al(n)) cos urT + + a(n)n+l cos (n + 1)o-T,

R CO f) + (d2 f8n+ + c1+ C c * + (nn+l cos (n + I) fr, Q(n) bl(n) sin or + * * * + bn"n+, sin (n + I)ofr,

The complementary functions of (34, a and b) and the terms arising from 2C1(2) in (34 b) are

pn = o(B1 (n)ea + B2(n)eWiar) + 2m (1 -

rn -B (n) eiTar + B2 (n)Me2 2+C2 (n)

The symbolic expressions for the particular integrals are the same as (25) with the appropriate changes in subscripts and superscripts. As at the previous steps the coefficients of cos or in the numerators of these expressions must vanish in order that pn and qn shall be periodic. We thus arrive at the two equations

(I _c2 + 3m2) (dl(n)8n + al(n)) + 3m( _2)/2(d2 Wan + ci(n)) =0, 3m,(1 _M2)l/2(di(n) 8n + a,i()) + [ =.2 + 3(1 _2)] (d (n)8 + e (n)) o

Since the functional determinant in these equations vanishes, the two equations are equivalent and can be satisfied by solving either for 8n. Thus

_(_ 2 + 1 + 3m2)al(n)- 3m (1-_m2)'/2 Cl(n) n -(_ff2 + t

+ 3m2)di(f)+ 3m(1_m2)12/d2(d2

With this choice of 8n the particular integrals will be periodic and will have the form

Pn - (of) + 2 (n) cos 2-r + * + ((n) n+l cos(n + or,

rn yo(n) + y2(n) cos 2or +* + -yn)n+1 cos(n + l)ff-,

On substituting the complete solution for rn in (34 c) and integrating we obtain




qn (2i/oa) (Bj(n)eia7 - B2(n)e-aT) - (3C1(n) + 2y (n))T

n+1 + C2Q(n) +? 83v (n sin vari,

v=2

and in order that this solution shall be periodic we must put

C -(n) (2/3).yo(n).

When the initial conditions are applied we obtain

C2=n) 0, Bi(n) - B2() - a constant.

Hence pn, qn and rn have the same form as (33) when j = n. This completes the induction. The construction of the solutions can therefore be carried on to any desired degree of accuracy.

The two sets of solutions can be obtained by restoring the subscripts 1 or 2 to o) and a.

9. The Final Form of the Solutions. On substituting the various values for pj, qj, rj in (19) and the results in (9) we obtain

00 j+1 x m + E ( Y FP(J) cos vo)yJ+I,

j=O v=O

00 j+1 = 0 + i ( E Gv(J) sin voTr)y1j+,

j=O v=1

oo j+1 t (1 _m2)V2 + E ( 2 HvD) COS voT) Y+1'

j=O v-=0 00

T (1+ E 8j7j) -:'(t -to). j=l

In the abo:ve equations m, 'y and to are the only parameters which remain arbitrary; m denoting the scale factor of the circular orbits, y that of the periodic oscillations near these orbits, and to the epoch. By substituting for 1 and 6 in the equations

y = Xcos (t- to) + e sin (t -to), z =Xsin (t - to) + e cos (t -to)

we may obtain the corresponding values of y and z. There are two sets of values, xl, yl, z1; X2, Y2, Z2, corresponding to the two electrons, but they are not independent inasmuch as the restrictions (2) hold

Xl=- - X2, Yl =- Y2.nZ 51-- 2-

10. Numerical Example. Mr. Smith assigned the, values

k'2 == .5 m .5, y .05, to-0,




and on completing the integrations up to the terms in p2, q2 and r2 he obtained

p = - .0025 + .064 cos 5r + .025 cos 2or - .002 cos 3ar, q = -.19 sinaT + .03 sin 2oT-.0003 sin 3ar, r= .0043 + .077 cos uT -.017 cos 2oT- .00027 cos 30T.

Using the subscript 1 on and a he found

=l .825, P1 = 127r/5, nearly. Values of t were then taken at approximately 300 intervals as t ranges from 0? to 211600, that is, through the complete period, and the numerical values of xi, yi, and z1 were computed. The values obtained near the beginning and near the end of the period are found in the accompanying Table.

to Xl y1 zi 0 .560 .00 .95

30 .555 .54 .77 60 .536 .87 .33 90 .515 .87 -.19

120 .488 .64 - .59 150 .464 .26 - .78 180 .450 - .10 -.80 210 .440 - .43 -.66 240 .475 - .65 -.46 270 .458 -.81 -.14 300 .480 - .81 .26 330 .506 - .60 .65 360 .530 -.17 .83

1800 .530 .17 .93 1836 .500 .68 .57 1890 .458 .81 - .14 1926 .443 .60 -.51 1980 .448 .10 - .80 2016 .470 - .36 -.76 2070 .515 -.87 -.19 2106 .542 - .81 .43 2160 .560 0 .95

A check was made on the work by making use of the vis viva integral (4 a). Various sets of computed values for xi, yi, z1 and their derivatives were used and the constant in the vis viva integral was found to range from 2.17 to 2.31.

The accompanying diagrams give the projections of the oscillations on the coordinate planes. The circular orbit is not shown in Fig. 2. Its projections in Fig. 1 and Fig. 3 are the y- and z-axes respectively.




11. Two-Dimensional Orbits. Two-dimensional pe'riodic oscillations near the circular orbits can be readily found by neglecting tho terms in x in the preceding construction. These orbits are coplanar with the circular orbits.

x+

C,rcu.'f/

___ ~~~~~~~~~~~~~~~~~~~~~~~~.....-......... . . . ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ V

Figl~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~A

/~~~~~~~~~~~~~~~~/

Fig.2 Fig 3 The actual construction was carried out but as no peculiarities were found it- is omitted. Mr. Smith computed an orbit and found curves similar to those in Fig. 2.

THE UNIVERSITY OF BRITISH COLUMBIA, VANCOUVER, CANADA.

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Periodic Orbits in the Problem of Three Bodies with Repulsive and Attractive Forces

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